Liquid water has several anomalous properties, including a non-monotonous dependence of density with temperature and an increase of thermodynamic response functions upon supercooling. Four thermodynamic scenarios have been proposed to explain the anomalies of water, but it is not yet possible to decide between them from experiments because of the crystallization and cavitation of metastable liquid water. Molecular simulations provide a versatile tool to study the anomalies and phase behavior of water, assess their agreement with the phenomenology of water under conditions accessible to experiments, and provide insight into the behavior of water in regions that are challenging to probe in the laboratory. Here we investigate the behavior of the computationally efficient monatomic water models mW and mTIP4P/2005REM, with the aim of unraveling the relationships between the lines of density extrema in the p-T plane, and the lines of melting, liquid-vapor spinodal and non-equilibrium crystallization and cavitation. We focus particularly on the conditions for which the line of density maxima (LDM) in the liquid emerges and disappears as the pressure is increased. We find that these models present a retracing LDM, same as previously found for atomistic water models and models of other tetrahedral liquids. The low-pressure end of the LDM occurs near the pressure of maximum of the melting line, a feature that seems to be general to models that produce tetrahedrally coordinated crystals. We find that the mW water model qualitatively reproduces several key properties of real water: (i) the LDM is terminated by cavitation at low pressures and by crystallization of ice Ih at high pressures, (ii) the LDM meets the crystallization line close to the crossover in crystallization from ice Ih to a non-tetrahedral four-coordinated crystal, and (iii) the density of the liquid at the crossover in crystallization from ice Ih to a four-coordinated non-tetrahedral crystal coincides with the locus of maximum in diffusivity as a function of pressure. The similarities in equilibrium and non-equilibrium phase behavior between the mW model and real water provide support to the quest to find a compressibility extremum, and determine whether it presents a maximum, in the doubly metastable region.

Water displays several anomalous thermodynamic, dynamic, and structural properties.1–5 For example, the density of liquid water reaches a maximum at 4 °C at ambient pressure and decreases upon further cooling. In the supercooled region, there is a rapid increase of response functions (such as the isothermal compressibility κT and isobaric specific heat capacity Cp) upon cooling1,4 that does not occur in “normal” liquids. Four hypotheses have been proposed to explain these anomalies of water. The stability limit4,6 and critical-point-free7–9 scenarios predict a monotonic line of density maxima (LDM) and a divergence of response functions on approaching the spinodal of liquid water in the deeply supercooled region, while the liquid-liquid critical point (LLCP)10 and singularity-free11 scenarios predict that the LDM would be reentrant and response functions would either diverge at the critical point in the LLCP scenario or go through maxima in the deeply supercooled region.

Experiments have not been able to decide between these possible scenarios. The hypothesized peaks or divergence of response functions lie in the so-called “no man’s land,” a region where homogeneous crystallization of supercooled liquid water is too fast to allow the study of the liquid. In a quest to determine whether the LDM of water retraces at negative pressures or monotonously approaches the liquid-vapor spinodal, the LDM has been determined in experiments from ˜1400 bars, at which it intersects the line of homogeneous crystallization of hexagonal ice (ice Ih), to about −1200 bars, below which cavitation of the liquid precludes further measurements.12,13 Cavitation occurs above the spinodal limit and effectively sets the low-pressure boundary of metastastable liquid water. The line of density maxima of water shows a monotonic behavior with a negative slope in all the range measured; it is not known whether at pressures below −1200 bars the temperature of maximum density of water monotonically increases towards the spinodal or turns around and retraces to lower temperatures as predicted by the LLCP and singularity free scenarios.

Molecular simulations provide a versatile tool to study the anomalies and phase behavior of water. Cavitation may be avoided by choosing a very small simulation cell, which is the case for most of the simulations of water. Using small simulation cells, the LDMs of the atomistic models of water such as ST2,10 TIP4P,14 SPC/E,15 and TIP4P/200516 were found to display reentrant behavior without the interference of cavitation. Those results are consistent with the predictions of the LLCP10 or singularity free11 scenarios, but not with the stability limit4 and critical-point-free7–9 ones. The crystallization of atomistic models of water is slow, making it possible to study the liquid under deeply supercooled conditions. For example, the ST2 water model has been shown to have a liquid-liquid transition (LLT) and maxima of response functions without the interference of crystallization,17,18 which is consistent with the LLCP scenario. The difficulty in crystallization, however, hinders the use of atomistic models in the elucidation of the interplay between crystallization and anomalies in water.

Coarse-grained models of water that produce tetrahedral order, such as the mW and mXREM models,19–22 spontaneously crystallize in the deeply supercooled region, which allows for the characterization of the homogeneous crystallization of water and its relation to the line of density maxima. The mW and mXREM water models21,22 represent each water molecule by one particle interacting through the short-range two- and three-body interactions modeled with the Stillinger-Weber (SW) potential.23 The mW model21 was parameterized using a “top-down”24 scheme by matching experimental observables, while the mXREM models22 were parameterized based on atomistic water models X (X = TIP4P/2005,16 TIP4P-Ew,25 SPC/E,15 and TIP3P26) using the “bottom-up”24 relative entropy minimization (REM)27–29 approach. mTIP4P/2005REM is the model that best reproduces the experimental properties of water among mXREM models, although in that regard it ranks lower than mW.22 These coarse-grained models have shown abilities to produce the characteristic anomalies and multiple key properties of liquid water,19–22,30–35 at ∼1% of the computational cost of atomistic models with electrostatic interactions. In particular, the mW model has been widely used to study the thermodynamics,21,31,32,34,36–39 crystallization,19,20,34,35,39–66 evaporation,67–72 anomalies,21,30–32,37,73,74 and glass transitions33,75 of water.

In this work, we investigate the relationships between the density anomaly, melting line, locus of the temperature of maximum crystallization rate Tx, cavitation line, and liquid-vapor spinodal in the mW and mTIP4P/2005REM monatomic models of water. We compare the relations in these two coarse-grained models with those in experiments of water,3,21,31,76–83 atomistic water models,10,14–16,25,26 and models of other tetrahedral liquids,22,30,74,84–87 with the goal of finding general correlations between the properties and assessing the ability of the mW and mTIP4P/2005REM models to reproduce the phenomenology of water.

The coarse-grained water models are evolved with molecular dynamics simulations performed in LAMMPS.88 The equations of motion are integrated using the velocity Verlet algorithm with time step 6 and 2 fs for the mW and mTIP4P/2005REM models, respectively, which conserves the energy in microcanonical simulations for all conditions of this study. All systems are simulated with periodic boundary conditions. Temperature and pressure are controlled with the Nose-Hoover thermostat and barostat with relaxation times of 100 and 1000 times the time step, respectively. The equilibrium melting temperaturesTm of hexagonal ice (ice Ih) as a function of pressure are determined from ice-liquid phase coexistence in the NpT ensemble.89 The two-phase systems consist of 9216 water particles, half in a slab of ice in contact with a slab of liquid. The uncertainty in Tm, ±0.5 K, is computed from the gap between the highest temperature at which ice grows and the lowest temperature at which it melts. The temperature of maximum crystallization rate, Tx, is determined from isobaric simulations for a cell with 8000 water particles cooled at constant rates from 0.01 to 1 K ns−1, following the protocols of Ref. 19. The cavitation pressures are calculated by isothermally decreasing the pressure of periodic cells with 8000 particles at a constant rate of 10.1325 bars/ns, until cavitation occurs. The line of density maxima (LDM) above the cavitation pressures is determined by performing isobaric cooling simulations at the rates of 1 K/ns for the systems with the cell of 8000 water molecules; that cooling rate is slow enough to allow equilibration of the liquid at each temperature for which we report a density extremum. To avoid cavitation and access to the LDM of the mW model at very negative pressures we perform isochoric simulations in a smaller system containing only 400 water particles. To accurately locate the points of the temperature of maximum density near the ends of the LDM for the mTIP4P/2005REM model, the two points at the low-pressure end and the two points at the high-pressure end of the LDM are computed from the polynomial fitting of the isochores of systems containing 8000 water molecules with 200 ns and 27 000 water molecules with 80 ns, respectively. The locus of the spinodal of the liquid with respect to vapor for the mW model is estimated by fitting the simulation pressures corresponding to number densities ranging from 0.0303 to 0.0327 (every 0.0002) at 240, 260, and 275 K with the equation of state4 p = p s 1 B ρ / ρ s 1 2 .

The phase diagram for water has been determined at positive pressures in experiments. The tetrahedrally coordinated ice Ih is the stable crystal, up to 2099 bars, where a non-tetrahedral but also four-coordinated crystal (ice III) becomes more stable.90,91 The triple point between ice Ih, liquid, and ice III is at 251.165 K and 2099 bars.90,91 Ice Ih is the stable crystal of the mW model at positive pressures, up to 11 870 bars where the sc16 crystal becomes more stable92 (Figure 1(a)). The ice Ih-liquid-sc16 triple point of mW is at 11 870 bars and 244 K.92 The high-pressure sc16 polymorph, though different from ice III, is also a non-tetrahedral but four-coordinated crystal. For the mTIP4P/2005REM model, sc16 becomes more stable than ice Ih at pressures above ∼6000 bars (Figure 1(b)). The sc16 crystal of mTIP4P/2005REM is less dense than the liquid, which results in a negatively slopped melting line. That is not the case for real water and mW water, for which the high-pressure crystalline phases (ice III in real water, sc16 in mW) are denser than the liquid.

FIG. 1.

Lines of the density maxima (red), temperature of maximum crystallization rate Tx (green), cavitation (blue), liquid-vapor spinodal (brown), and coexistence between ice Ih and liquid (black), sc16 and liquid (cyan), and ice Ih and sc16 phases (orange) for the mW model (panel (a)) and mTIP4P/2005REM model (panel (b)). Tm denotes the melting line of ice Ih. Solid coexistence lines for mW were computed from free energy calculations in Ref. 92, while black and turquoise triangles are from two-phase coexistence in this work. Other solid lines in panel (a) and (b) are guide to the eyes. The glass transition temperature Tg (purple diamond) in panel (b) denotes the temperature at which the diffusion coefficient of mTIP4P/2005REM is lower than 10−8 cm2 s−1 and the liquid cannot be properly equilibrated in the simulations. The shaded grey rectangle in panel (b) denotes the area where the LDM cannot be resolved because of too small variation of pressure along isochores. The solid black circle (TmD) in panel (b) indicates a density minimum of the mTIP4P/2005REM model determined from polynomial fitting of the isochores of the 8000-molecule systems. The loci of the equilibrium and non-equilibrium lines are sensitive to the time step used in the simulations and are converged for the values used in the present study. The magenta dashed line in panel (b) is guide to the eyes, to show that the pressure of the low-pressure end of the LDM is close to the pressure where the slope of the melting line changes the sign. For the cavitation points in both models, the solid blue up-triangles represent the cavitation of the liquid, while the open blue up-triangles are the cavitation from a non-equilibrated amorphous state. Cavitation precludes us from locating the exact low-pressure end of the LDM in the mW water model.

FIG. 1.

Lines of the density maxima (red), temperature of maximum crystallization rate Tx (green), cavitation (blue), liquid-vapor spinodal (brown), and coexistence between ice Ih and liquid (black), sc16 and liquid (cyan), and ice Ih and sc16 phases (orange) for the mW model (panel (a)) and mTIP4P/2005REM model (panel (b)). Tm denotes the melting line of ice Ih. Solid coexistence lines for mW were computed from free energy calculations in Ref. 92, while black and turquoise triangles are from two-phase coexistence in this work. Other solid lines in panel (a) and (b) are guide to the eyes. The glass transition temperature Tg (purple diamond) in panel (b) denotes the temperature at which the diffusion coefficient of mTIP4P/2005REM is lower than 10−8 cm2 s−1 and the liquid cannot be properly equilibrated in the simulations. The shaded grey rectangle in panel (b) denotes the area where the LDM cannot be resolved because of too small variation of pressure along isochores. The solid black circle (TmD) in panel (b) indicates a density minimum of the mTIP4P/2005REM model determined from polynomial fitting of the isochores of the 8000-molecule systems. The loci of the equilibrium and non-equilibrium lines are sensitive to the time step used in the simulations and are converged for the values used in the present study. The magenta dashed line in panel (b) is guide to the eyes, to show that the pressure of the low-pressure end of the LDM is close to the pressure where the slope of the melting line changes the sign. For the cavitation points in both models, the solid blue up-triangles represent the cavitation of the liquid, while the open blue up-triangles are the cavitation from a non-equilibrated amorphous state. Cavitation precludes us from locating the exact low-pressure end of the LDM in the mW water model.

Close modal

Water can remain in a supercooled liquid state at temperatures below the melting point until homogeneous nucleation of ice occurs. The crystal structure of ice nucleated from supercooled water in experiments crossovers from ice I to ice III when the density of the supercooled liquid is 1.12 g cm−3, at essentially the same pressure of the ice I-liquid-ice III triple point. Similarly, for mW water there is a nucleation crossover from ice Ih to sc16 at a density of 1.13 g cm−3. Because of the lower compressibility of mW compared to real water,21 the pressure (∼9000 bars) corresponding to this density is significantly higher than in experiments, and about 3000 bars below the ice Ih-liquid-sc16 triple point (Figure 1(a)). Supercooled mTIP4P/2005REM did not crystallize at pressures above ∼5000 bars in simulations at cooling rates as low as 0.01 K ns−1.

Interestingly, the nucleation crossover from a tetrahedral to non-tetrahedral crystal in real3,76 and mW21 water coincides with the locus of the maximum in diffusivity as a function of pressure Dmax(p). These results point to a change from tetrahedral to non-tetrahedral order in the liquid at densities around 1.12 g cm−3. This is supported by a non-equilibrium transition between low-density amorphous ice (LDA) and high-density amorphous ice (HDA) in both real77–79 and mW33 water around that density.

The sign of the slope of the non-equilibrium crystallization line vs pressure, Tx(p), of the coarse-grained models follows that of the equilibrium melting line Tm(p) (Figure 1). Tx(p) of mTIP4P/2005REM water is almost parallel to the Tm(p), which displays a maximum Tmmax (at which the volume of liquid and crystal are identical) and retraces on further extension. Parallel Tx(p) and Tm(p) lines that retrace at positive pressure were also found for a monatomic isotropic model that stabilizes diamond.93 The crystallization line of the mW model is not parallel to the melting line. Same as in real water,80,81Tx(p) of mW has a more pronounced slope than Tm(p) at positive pressures (Figure 1(a)). Addition of salts also produces more change in Tx than in Tm for both mW52 and real water.82 Both Tm(p) and Tx(p) of mW meet the cavitation line at negative pressures before being able to retrace. We note that the guest-free sII clathrate becomes more stable than ice Ih for mW at pressures lower than about −1300 bars and temperatures lower than 275 K.65 However, liquid mW water cooled in the range of stability of the sII clathrate nucleates metastable ice Ih, because the structure of the supercooled liquid approaches the one of ice Ih on cooling.65 A Tmmax in the ice Ih-liquid equilibrium line at negative pressures has been proposed for real water,90,94,95 but no experiment has yet been able to find it. It is likely that, if cavitation were avoided, Tx(p) of mW and real water would also become more negatively slopped and, as observed for mTIP4P/2005REM (Figure 1(b)), may retrace to lower temperatures at negative pressures mirroring the expected behavior of the equilibrium melting line.

We now focus on the LDM in the pT plane for the coarse-grained mW and mTIP4P/2005REM models of water. The LDM of the coarse-grained models is always below the melting line, different from the results for real water for which the LDM crosses the melting line at a pressure of around 300 bars.80 The LDMs of the coarse-grained water models show a reentrant behavior (Figure 1). This is also the case for the other coarse-grained mXREM models and fully atomistic water models (e.g., ST2,10 TIP4P,14 SPC/E,15 and TIP4P/200516), as well as for models of other tetrahedral substances (Si, SiO2, and BeF230,84,85), for which the LDM line has been reported. The retracing LDM found for all these models is incompatible with the stability limit4 and critical-point-free7–9 scenarios that predict monotonous LDM, but is consistent with the predictions of the liquid-liquid critical point (LLCP)10 or singularity free11 scenarios. Indeed, some of these water models are known to have a LLCP (e.g., ST218) and others (e.g., mW31,39) are known to present anomalies without a liquid-liquid transition.

A retracing LDM cannot meet the spinodal of the liquid with respect to the vapor.10,11 In practice, the liquid cavitates before reaching the spinodal limit. Cavitation always occurs at pressures between the liquid-vapor equilibrium line and the spinodal. Figure 1 shows the cavitation lines for the mW and mTIP4P/2005REM water models computed from simulation cells with 8000 molecules. There is a significant difference between the two models. The LDM of mTIP4P/2005REM turns around and ends at positive pressure, p = 300 bars and T = 173 ± 5 K, where the liquid is supercooled but stable with respect to the vapor. The LDM of mW water, on the other hand, turns around and ends at negative pressures, where the liquid is doubly metastable with respect to crystallization and cavitation, a scenario also reported for the ST217 and TIP4P/200596–98 water models. The LDM of water in experiments has been determined down to about −1200 bars, where it meets the cavitation line.12,13 It is not known whether the LDM of real water meets the spinodal at lower pressures, or retraces to lower temperatures as it occurs for all molecular models of water.

Thermodynamic constraints require that the reentrant LDM must transform into a line of density minima (LDm) at low pressures.4,99,100 The LDm in coarse-grained (Figure 1) and atomistic17,101–105 water models, when accessible, occurs at temperatures where water is metastable with respect to ice. Ice formation can be prevented by nanoconfinement. Experiments of deeply cooled nanoconfined water have been interpreted to reveal a density minimum,106–110 although some authors have challenged that interpretation.107,108,111 Density minima have been identified in simulations of bulk water with atomistic water models,17,101–105 for which the kinetics of crystallization is slow compared to the relaxation time of the liquid. Ice formation is relatively fast in the coarse-grained mXREM and mW water models.19,21,22 We investigate whether the line of density maxima of the monatomic water models transitions to a line of density minima before fast crystallization closes the window for studying the liquid state. We find that the LDM of mTIP4P/2005REM is enclosed within the region where the liquid is supercooled but more stable than the vapor. For that model we identify a density minimum at p = 354 bars T = 165 ± 2 K (Figure 1(b)), slightly above Tx = 159 ± 3 K. Crystallization prevents the study of the LDm of mTIP4P/2005REM at higher pressures. For mW water we cannot observe the transition from LDM to LDm before cavitation results in the demise of the doubly metastable liquid state. Similarly, cavitation—not crystallization—terminates the experimental exploration of the LDM of real water at negative pressures.

The singularity free11 and LLCP10 scenarios predict that thermodynamic response functions would reach a maximum in deeply supercooled water. Maxima in Cp and κT have been found in the ST217 and TIP4P/200596–98 models. Experiments at positive pressures have demonstrated a fast rise in Cp and κT when liquid water is cooled towards the homogeneous nucleation temperature,1,2,80 but crystallization prevented a verification of the existence of maxima in these response functions. It has been proposed that in the doubly metastable region of real water, the temperature of maximum κT may lie above the temperature of homogeneous nucleation.12,112,113 In such case, and if cavitation could be avoided, it would be possible to determine whether the compressibility of liquid water indeed presents a maximum, a distinct feature that would help decide between thermodynamic scenarios proposed to explain the origin of anomalies in water. For mW water, κT of the liquid increases on cooling but does not achieve a maximum at positive pressures before crystallization becomes unavoidable,31 same as in real water.1,2,80 There has not yet been a characterization of κT of mW at negative pressures. Thermodynamic constraints require that if response functions have maxima, the line of κT extrema must pass through the point of maximum temperature on the LDM.11 Because of the gap between the maximum in the LDM and Tx(p) and the retracing behavior of the crystallization line at low pressures displayed by the mTIP4P/2005REM and mW models (Figure 1), it is possible that the locus of κT extrema in these models would emerge above Tx and could be measured without interference from cavitation. The similarities in behavior of the mW model and real water add support to the quest12,113 to find a κT extremum, and determine whether it corresponds to a maximum, in the doubly metastable region of real water.

We now focus on the low-pressure end of the LDM, at which the slope of the LDM vs temperature becomes zero.17,100 Since liquid water is more compressible than ice Ih, the density of the liquid approaches the one of the crystal on expansion, until the liquid becomes the less dense phase. The LDM in the coarse-grained mTIP4P/2005REM model terminates when the liquid becomes less dense than the crystal. This termination corresponds to a pressure slightly below the one of the Tmmax (Figure 1(b)), where the volume of liquid and crystal is identical. The small gap in pressure between the low-pressure end of the LDM and the Tmmax originates on the different thermal expansivities of water and ice as they are cooled from Tm to LDM. For the mW model the line of density anomalies comes close to the liquid-vapor spinodal and cavitation occurs (even for simulations with as little as 400 molecules) before we can observe the LDM vanish. Cavitation also precludes the location of the end of the LDM for real water,12,13,114,115 for which a Tmmax at negative pressures has been proposed but is not experimentally accessible.90,94,95 Using data from Refs. 17 and 116 for ST2 and Refs. 30, 96, 97, 117, and 118 for TIP4P/2005, we find that the LDM of these atomistic water models at low pressures also ends slightly below the pressure at which the slope of the melting line changes sign. Same correlation is found for a two-body potential that stabilizes diamond93 and for a family of Stillinger-Weber potentials,74 suggesting that the coincidence of the pressure for which the LDM ends on extension and the melting temperature of the tetrahedral crystal reaches a maximum is a general feature of models that produce tetrahedrally coordinated crystals.

At high positive pressures, the LDM of water intersects the non-equilibrium crystallization line of ice Ih in both experiments80,83 and the mW model (Figure 1(a)). In both cases the LDM ends below the region where there is a crossover to a non-tetrahedral crystal. Equivalent behavior was also reported for a family of Stillinger-Weber monatomic tetrahedral liquids at constant pressure as a function of strength of tetrahedral order in the potential.74 The LDM of the mTIP4P/2005REM model tends to vanish at 1.11 g cm−3 (p ≈ 6200 bars), above which we cannot resolve density extrema in the simulation data (Figure 1(b)). mTIP4P/2005REM liquid did not crystallize in our simulations above 5000 bars (see Section III A), hence no data is available for the crossover in crystallization from ice I to sc16. The LDMs of atomistic models ST217 and TIP4P/200596,97 have been computed to high-pressure end. The LDM of SPC/E has also been computed to high pressures,76,119 though not yet to the end of the LDM, where its slope in the pT plane should become zero. Different from the case of real water, mW and TIP4P/2005, the LDMs of ST217,116 and SPC/E76,118 end in the region of stability of ice VII and ice II, respectively. Nevertheless, the high pressure end of the LDM for real water3,76 and mW (Figure 1(a)), ST2,17,116 TIP4P/2005,96,97,118,120 and SPC/E76,119 water models always occurs at a density lower than the density of the locus of the maximum in diffusivity as a function of pressure Dmax(p). As the liquid density at Dmax(p) coincides with the density of crossover in crystallization from tetrahedral ice Ih to a non-tetrahedral crystal for real and mW water (Section III A), it would be interesting to investigate the relation between the high-pressure end of the LDM in the ST2 and SPC/E model, for which the LDM ends in the region of stability of ice VII and ice II, respectively, and the locus of the crossover in crystallization from tetrahedral ice Ih to a non-tetrahedral crystal in these water models.

In this work we address the relationships between the density maximum/minimum, the melting and spinodal lines and the non-equilibrium crystallization and cavitation lines in the mW and mTIP4P/2005REM monatomic coarse-grained models of water. The line of density maxima (LDM) in these water models displays reentrant behavior, same as previously reported for atomistic models of water10,14–16 and molecular models of other tetrahedral liquids.30,84,85 These results are consistent with either the singularity-free or liquid-liquid critical point scenarios, but not with the critical point-free and stability limit ones. A monotonic LDM has been theoretically predicted for some lattice models of water-like fluids,6,121–125 but to our knowledge not yet found in molecular simulations with a water model. However, a very recent ab initio molecular dynamics study of liquid Si suggests, but cannot conclusively prove, a phase behavior consistent with the critical-point-free scenario.126 It is still an open question what are the interactions needed to produce a liquid in which the temperature of maximum density increases monotonously with decreasing pressure until it intersects the spinodal.

We find that at high positive pressures the crossover in nucleation from tetrahedral to non-tetrahedral crystals coincides with the locus of the maximum in diffusivity as a function of pressure Dmax(p) in the mW model, same as in experiments. Though at quite different pressures, the density of the crossover in the mW model is very close to that in experiments. These results point to a change from tetrahedral to non-tetrahedral order when the liquid achieves a density around 1.12 g cm−3 for real and mW water. This is supported by a non-equilibrium transition around that density between low-density amorphous ice (LDA) and high-density amorphous ice (HDA) in both real77–79 and mW33 water.

There is a significant difference between the two coarse-grained water models: the low-pressure end of the LDM in the mTIP4P/2005REM model is located in the supercooled region but stable with respect to the vapor, while in the mW model the LDM crosses the cavitation line and ends at pressures close to the liquid-vapor spinodal. We show that the low-pressure boundary of the LDM occurs near the pressure of maximum melting temperature: as soon as the liquid becomes less dense than the crystal, the density anomaly disappears and normal liquid behavior is recovered. This correlation seems to be generic to all models that produce tetrahedrally coordinated crystals. The homogeneous crystallization line Tx(p) becomes more negative as the pressure decreases and tends to retrace to lower temperatures. The relative positions of the LDM and crystallization line in the coarse-grained water models add support to the quest to experimentally measure compressibility extrema and assess the existence of a maximum in the doubly metastable region of water. Such experiments would help to elucidate the origin of water anomalies.

We find that although the mW water model does not quantitatively reproduce the experimental phase diagram of water, it qualitatively mimics real water in several key properties: (i) the LDM meets the cavitation line at low pressures and the ice Ih crystallization line at high pressures; (ii) the pressure at which the LDM meets the crystallization line is close to the crossover in crystallization from ice Ih to a non-tetrahedral four-coordinated crystal; (iii) the density of liquid at the crystallization crossover is ∼1.12 g cm−3, and (iv) the crossover coincides with the locus of the maximum in diffusivity as a function of pressure.

The similarities in equilibrium and non-equilibrium phase behavior between the mW model and real water add interest in exploring the mW model in the doubly metastable and high positive pressure regions, and in re-engineer the interactions in the monatomic water model to reproduce the structures of the high density phases of water.

We gratefully acknowledge Hajime Tanaka for kindly sharing with us the coexistence lines of mW water computed in Ref. 92, and Mikhail A. Anisimov for helpful discussions and insightful comments. This research was sponsored by the Army Research Laboratory under Cooperative Agreement No. W911NF-12-2-0023. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. We thank the Center of High Performance Computing at The University of Utah for technical support and an award of computer time.

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